Tài liệu Đề tài " Deligne’s conjecture on 1-motives " ppt

Saito Abstract We reformulate a conjecture of Deligne on 1-motives by using the integralweight filtration of Gillet and Soul´e on cohomology, and prove it.. If the degree of cohomology is

Deligne’s conjecture on

1-motives

By L Barbieri-Viale, A Rosenschon, and M Saito

Deligne’s conjecture on 1-motives

By L Barbieri-Viale, A Rosenschon, and M Saito

Abstract

We reformulate a conjecture of Deligne on 1-motives by using the integralweight filtration of Gillet and Soul´e on cohomology, and prove it This impliesthe original conjecture up to isogeny If the degree of cohomology is at mosttwo, we can prove the conjecture for the Hodge realization without isogeny,and even for 1-motives with torsion

Let X be a complex algebraic variety We denote by H(1)j (X,Z) the imal mixed Hodge structure of type {(0, 0), (0, 1), (1, 0), (1, 1)} contained in

max-H j (X, Z) Let H j

(1)(X,Z)fr be the quotient of H(1)j (X,Z) by the torsion group P Deligne ([10, 10.4.1]) conjectured that the 1-motive corresponding

sub-to H(1)j (X,Z)fr admits a purely algebraic description, that is, there should

ex-ist a 1-motive M j (X)fr which is defined without using the associated analytic

space, and whose image r H (M j (X)fr) under the Hodge realization functor r H

(see loc cit and (1.5) below) is canonically isomorphic to H(1)j (X,Z)fr(1) (and

similarly for the l-adic and de Rham realizations).

This conjecture has been proved for curves [10], for the second cohomology

of projective surfaces [9], and for the first cohomology of any varieties [2] (seealso [25]) In general, a careful analysis of the weight spectral sequence in

Hodge theory leads us to a candidate for M j (X)frup to isogeny (see also [26]).However, since the torsion part cannot be handled by Hodge theory, it is arather difficult problem to solve the conjecture without isogeny

In this paper, we introduce the notion of an effective 1-motive which

ad-mits torsion By modifying morphisms, we can get an abelian category of1-motives which admit torsion, and prove that this is equivalent to the category

of graded-polarizable mixed Z-Hodge structures of the above type However,

our construction gives in general nonreduced effective 1-motives, that is, the

discrete part has torsion and its image in the semiabelian variety is nontrivial

1991 Mathematics Subject Classification.14C30, 32S35.

Key words and phrases 1-motive, weight filtration, Deligne cohomology, Picard group.

Let Y be a closed subvariety of X Using an appropriate ‘resolution’, we can define a canonical integral weight filtration W on the relative cohomology

H j (X, Y ; Z) This is due to Gillet and Soul´e ([14, 3.1.2]) if X is proper See also (2.3) below Let H(1)j (X, Y ;Z) be the maximal mixed Hodge structure of the

considered type contained in H j (X, Y ;Z) It has the induced weight filtration

W , and so do its torsion part H(1)j (X, Y ;Z)tor and its free part H(1)j (X, Y ;Z)fr.Using the same resolution as above, we construct the desired effective 1-motive

M j (X, Y ) In general, only its free part M j (X, Y )fris independent of the choice

of the resolution By a similar idea, we can construct the derived relativePicard groups together with an exact sequence similar to Bloch’s localizationsequence for higher Chow groups [7]; see (2.6) Our first main result shows

a close relation between the nonreduced structure of our 1-motive and theintegral weight filtration:

0.1 Theorem There exists a canonical isomorphism of mixed Hodge

structures

φfr: r H (M j (X, Y ))fr(−1) → W2H(1)j (X, Y ;Z)fr, such that the semiabelian part and the torus part of M j (X, Y ) correspond re-

spectively to W1H(1)j (X, Y ;Z)fr and W0H(1)j (X, Y ;Z)fr A quotient of its crete part by some torsion subgroup is isomorphic to Gr W2 H(1)j (X, Y ; Z) Fur-

dis-thermore, similar assertions hold for the l-adic and de Rham realizations.

This implies Deligne’s conjecture for the relative cohomology up to isogeny

As a corollary, the conjecture without isogeny is reduced to:

H(1)j (X, Y ;Z)fr= W2H(1)j (X, Y ;Z)fr.

This is satisfied if the GrW q H j (X, Y ; Z) are torsion-free for q > 2 The problem

here is that we cannot rule out the possibility of the contribution of the torsionpart of GrW q H j (X, Y ; Z) to H j

(1)(X, Y ;Z)fr By construction, M j (X, Y ) does not have information on W1H(1)j (X, Y ;Z)tor, and the morphism φfr in (0.1) isactually induced by a morphism of mixed Hodge structures

The proof of these theorems makes use of a cofiltration on a complex

of varieties, which approximates the weight filtration, and simplifies manyarguments The key point in the proof is the comparison of the extensionclasses associated with a 1-motive and a mixed Hodge structure, as indicated

in Carlson’s paper [9] This is also the point which is not very clear in [26]

We solve this problem by using the theory of mixed Hodge complexes due

to Deligne [10] and Beilinson [4] For the comparison of algebraic structures

on the Picard group, we use the theory of admissible normal functions [29].This also shows the representability of the Picard type functor However, for

an algebraic construction of the semiabelian part of the 1-motive M j (X, Y ),

we have to verify the representability in a purely algebraic way [2] (see also[26]) The proof of (0.2) uses the weight spectral sequence [10] with integralcoefficients, which is associated to the above resolution; see (4.4) It is theneasy to show

0.3 Proposition Deligne’s conjecture without isogeny is true if E p,j −p

The paper is organized as follows In Section 1 we review the theory

of 1-motives with torsion In Section 2, the existence of a canonical integralfiltration is deduced from [17] by using a complex of varieties (See also [14].)

In Section 3, we construct the desired 1-motive by using a cofiltration on a

complex of varieties, and show the compatibility for the l-adic and de Rham

realizations After reviewing mixed Hodge theory in Section 4, we prove themain theorems in Section 5

Acknowledgements The first and second authors would like to thank the

European community Training and Mobility of Researchers Network titled

Algebraic K -Theory, Linear Algebraic Groups and Related Structures for

financial support

Notation In this paper, a variety means a separated reduced scheme of

finite type over a field

1 1-Motives

We explain the theory of 1-motives with torsion by modifying slightly [10].This would be known to some specialists

1.1 Let k be a field of characteristic zero, and k an algebraic closure of k.

(The argument in the positive characteristic case is more complicated due tothe nonreduced part of finite commutative group schemes; see [22].)

An effective 1-motive M = [Γ → G] over k consists of a locally finite f

commutative group scheme Γ/k and a semiabelian variety G/k together with

a morphism of k-group schemes f : Γ → G such that Γ(k) is a finitely generated

abelian group Note that Γ is identified with Γ(k) endowed with Galois action because k is a perfect field Sometimes an effective 1-motive is simply called

a 1-motive, since the category of 1-motives will be defined by modifying only

morphisms A locally finite commutative group scheme Γ/k and a semiabelian variety G/k are identified with 1-motives [Γ → 0] and [0 → G] respectively.

An effective morphism of 1-motives

u = (ulf, usa) : M = [Γ → G] → M f = [Γ f → G
]

consists of morphisms of k-group schemes ulf : Γ → Γ  and usa : G → G 

forming a commutative diagram (together with f, f ) We will denote by

Homeff(M, M )

the abelian group of effective morphisms of 1-motives

An effective morphism u = (ulf, usa) is called strict, if the kernel of usa is

connected We say that u is a quasi-isomorphism if usa is an isogeny and if wehave a commutative diagram with exact rows

(i.e if the right half of the diagram is cartesian)

We define morphisms of 1-motives by inverting quasi-isomorphisms from

the right; i.e a morphism is represented by u ◦ v −1 with v a quasi-isomorphism.

More precisely, we define

(1.1.2) Hom(M, M ) = lim

−→Homeff( M , M  ),

where the inductive limit is taken over isogenies G → G, and  M = [Γ →  G]

with Γ = Γ× G G (This is similar to the localization of a triangulated category

in [33].) Here we may restrict to isogenies n : G → G for positive integers n,

because they form a cofinal index subset Note that the transition morphisms

of the inductive system are injective by the surjectivity of isogenies togetherwith the property of fiber product By (1.2) below, 1-motives form a categorywhich will be denoted byM1(k).

Let Γtor denote the torsion part of Γ, and put Mtor= Γtor∩ Ker f This is

identified with [Mtor → 0], and is called the torsion part of M We say that M

is reduced if f (Γtor) = 0, torsion-free if Mtor = 0, free if Γtor = 0, and torsion

if Γ is torsion and G = 0 (i.e if M = Mtor) Note that M is free if and only if

it is reduced and torsion-free We say that M has split torsion, if Mtor ⊂ Γtor

is a direct factor of Γtor

We define Mfr= [Γ/Γtor → G/f(Γtor)] This is free, and is called the free

part of M If M is torsion-free, Mfr is naturally quasi-isomorphic to M This implies that [Γ/Mtor → G] is quasi-isomorphic to Mfr in general, and (1.3)gives a short exact sequence

0→ Mtor→ M → Mfr→ 0.

Remark If M is free, M is a 1-motive in the sense of Deligne [10] We

can show

(1.1.3) Homeff(M, M  ) = Hom(M, M )

for M, M  ∈ M1(k) such that M  is free This is verified by applying (1.1.1)

to the isogenies G → G in (1.1.2) In particular, the category of Deligne

1-motives, denoted byM1(k)fr, is a full subcategory ofM1(k) The functoriality

of M → Mfr implies

(1.1.4) Hom(Mfr, M  ) = Hom(M, M )

for M ∈ M1(k), M  ∈ M1(k)fr In other words, the functor M → Mfr is leftadjoint of the natural functorM1(k)fr→ M1(k).

1.2 Lemma For any effective morphism u :  M → M  and any quasi

-isomorphism  M  → M  , there exists a quasi -isomorphism  M →  M together with a morphism v :  M →  M  forming a commutative diagram Furthermore,

v is uniquely determined by the other morphisms and the commutativity In particular, we have a well -defined composition of morphisms of 1-motives (as

in [33])

(1.2.1) Hom(M, M )× Hom(M  , M )→ Hom(M, M  ).

Proof For the existence of  M , it is sufficient to consider the semiabelian

part G by the property of fiber product Then it is clear, because the isogeny

n : G  → G  factors through G  → G  for some positive integer n, and it is

enough to take n :  G →  G We have the uniqueness of v for  G since there is

no nontrivial morphism of G to the kernel of the isogeny  G  →  G which is a

torsion group The assertion for Γ follows from the property of fiber product.Then the first two assertions imply (1.2.1) using the injectivity of the transitionmorphisms

1.3 Proposition Let u : M → M  be an effective morphism of

1-motives Then there exists a quasi -isomorphism  M  → M  such that u is

lifted to a strict morphism u  : M →  M  (i.e Ker u 

sa is connected ) In ular, M1(k) is an abelian category.

partic-Proof It is enough to show the following assertion for the semiabelian

variety part: There exists an isogeny G  → G  with a morphism u 

sa : G →  G 

lifting usasuch that Ker u 

sais connected (Indeed, the first assertion implies theexistence of kernel and cokernel, and their independence of the representative

of a morphism is easy.)

For the proof of the assertion, we may assume that Ker usa is torsion,

dividing G by the identity component of Ker usa Let n be a positive integer annihilating E := Ker usa (i.e E ⊂ n G) We have a commutative diagram

Let G  be the quotient of G  by usaι( n G), and let q : G  →  G  denote the

projection Since usaι( n G) ⊂ ι (n G  ), there is a canonical morphism q  :



G  → G  such that q  q = n : G  → G  Then the usa in the right column

of the diagram is lifted to a morphism u 

sa: G →  G  (whose composition with

q  coincides with usa), because G is identified with the quotient of G by n G.

Furthermore, Im u 

sa is identified with the quotient of G by n G + E, and the

last term coincides withn G by the assumption on n Thus u 

sa is injective, andthe assertion follows

Remark An isogeny of semiabelian varieties G  → G with kernel E

cor-responds to an injective morphism of 1-motives

[0→ G ]→ [E → G ] = [0→ G].

1.4 Lemma Assume k is algebraically closed Then, for a 1-motive M ,

there exists a quasi -isomorphism M  → M such that M = [Γ f → G
 ] has split

torsion.

Proof Let n be a positive integer such that E := Γtor∩Ker f is annihilated

by n Then G  is given by G with isogeny G  → G defined by the multiplication

by n Let Γ = Γ× G G  We have a diagram of the nine lemma

The l-primary torsion subgroup of G is identified with the quotient of V l G :=

T l G ⊗Zl Ql by M := T l G Let M  be the Zl -submodule of V l G such that

M  ⊃ M and M  /M is isomorphic to the l-primary part of f (Γtor) Then

there exists a basis {e i }1≤i≤r of M  together with integers c i(1≤ i ≤ r) such

that{l c i e i }1≤i≤r is a basis of M So the assertion is reduced to the following,

because the assumption on the second exact sequence

0→ n G → f (Γ

tor)→ f(Γtor)→ 0

is verified by the above argument

Sublemma Let 0 → A i → B i → C → 0 be short exact sequences of finite abelian groups for i = 1, 2 Put B = B1× C B2 Assume that the second exact sequence (i.e., for i = 2) is the direct sum of

0→ Z/nZ → Z/nb j Z → Z/b j Z → 0,

such that A1 is annihilated by n Then the projection B → B2 splits.

Proof We see that B corresponds to (e1, e2) ∈ Ext1(C, A1× A2), where

the e i ∈ Ext1(C, A i) are defined by the exact sequences Then it is enough to

construct a morphism u : A2 → A1 such that e1 is the composition of e2 and

u, because this implies an automorphism of A1× A2 over A2 which is defined

by (a1, a2) → (a1− u(a2), a2) so that (e1, e2) corresponds to (0, e2) (Indeed,

it induces an automorphism of B over B2 so that e1 becomes 0.) But the

existence of such u is clear by hypothesis This completes the proof of (1.4).

The following is a generalization of Deligne’s construction ([10, 10.1.3]).1.5 Proposition If k = C, we have an equivalence of categories (1.5.1) r H:M1(C)→ MHS ∼ 1,

where MHS1 is the category of mixed Z-Hodge structures H of type

{(0, 0), (0, −1), (−1, 0), (−1, −1)}

such that Gr W −1 HQ is polarizable.

Proof The argument is essentially the same as in [10] For a 1-motive

M = [Γ → G], let Lie G → G be the exponential map, and Γ f 1 be its kernel.Then we have a commutative diagram with exact rows

We get W −1 HQ from Γ1, and W −2 HQ from the corresponding exact sequence

for the torus part of G (See also Remark below.)

We can verify that HZ and F0 are independent of the representative of M

(i.e a quasi-isomorphism induces isomorphisms of HZ and F0) Indeed, for an

isogeny M  → M, we have a commutative diagram with exact rows

and the assertion follows by taking the base change by Γ→ G So we get the

canonical functor (1.5.1) We show that this is fully faithful and essentiallysurjective (To construct a quasi-inverse, we have to choose a splitting of the

torsion part of HZ for any H ∈ MHS1.)

For the proof of the essential surjectivity, we may assume that H is either

torsion-free or torsion Note that we may assume the same for 1-motives by

(1.4) But for these H we have a canonical quasi-inverse as in [10] Indeed, if

H is torsion-free, we lift the weight filtration W to HZ so that the GrW k HZ are

torsion-free Then we put

0 H → 0 It is easy to see that this is

a quasi-inverse The quasi-inverse for a torsion H is the obvious one.

As a corollary, we have the full faithfulness of r H for free 1-motives using

(1.1.3) So it remains to show that (1.5.1) induces

because Ext1(GrW0 M, M ) = Ext1(Γ, Γ ) = 0 Since we have the corresponding

exact sequence for mixed Hodge structures and the assertion for GrW0 M is

clear, we may assume M = W −1 M , i.e., Γ = 0.

Let T (G) denote the Tate module of G This is identified with the pletion of HZ using (1.5.3) Then

com-Hom(M, M  ) = Hom(T (G), Γ  ) = Hom(H

Z, Γ  ),

and the assertion follows

Remark Let T be the torus part of G Then we get in (1.5.2) the integral

2.1 Let V k denote the additive category of k-varieties, where a morphism

X  → X  is a (formal) finiteZ-linear combination i [f i ] with f i a morphism

of connected component of X  to X  It is identified with a cycle on X  × k X 

by taking the graph (This is similar to a construction in [14].) We say that amorphism 

i n i [f i ] is proper, if each f i is The category of k-varieties in the

usual sense is naturally viewed as a subcategory of the above category For a

k-variety X, we have similarly the additive category V X consisting of proper

k-varieties over X, where the morphisms are assumed to be defined over X in

the above definition

Since these are additive categories, we can define the categories of plexesC k , C X , and also the categories K k , K X where morphisms are considered

com-up to homotopy as in [33] We will denote an object of C X , K X (or C k , K k)

by (X • , d), where d : X j → X j −1 is the differential, and will be often omitted

to simplify the notation The structure morphism is denoted by π : X • → X.

(This lower index of X • is due to the fact that we consider only contravariant

functors from this category.) For i ∈ Z, we define the shift of complex by

(X • [i]) p = X p+i We say that Y • is a closed subcomplex of X • if the Y i are

closed subvarieties of X i, and are stable by the morphisms appearing in the

differential of X •

of smooth quasi-projective varieties (Here nsqp stands for nonsingular and

quasiprojective.) Let D be a closed subvariety of X We denote by C b

X Dnsqp

the full subcategory of C b

Xnsqp consisting of X • such that D j := π −1 (D) ∩

X j is locally either a connected component or a divisor with simple normal

crossings for any j Here simple means that the irreducible components of

D j are smooth For an integer j, let C b, ≥j

X denote the full subcategory of C b

X and a finite filtration G on X 

• such that the restriction of G to each component X 

j is given by direct factors, and for each integer i there exists

a birational proper morphism of k-varieties g : Y  → Y together with a closed

subvariety Z of Y satisfying the following condition: Letting Z  = (Y  × Y Z)red,

the morphism g : Y  \ Z  → Y \ Z is an isomorphism and the graded piece

up to a shift of complex Clearly, this condition is stable by mapping cone

We say that a morphism X 

• → X • inC b

X orK b

X is a strong quasi-isomorphism

if its mapping cone is strongly acyclic inK b

X This condition is stable by positions, using the octahedral axiom of the triangulated category Similarly,

com-if vu and u or v are strongly acyclic, then so is the remaining (It is not clear

whether the strongly acyclic complexes form a thick subcategory in the sense

with coefficient ±1 and f i has the lifting property We say that a morphism

u : X  → X in V k is of birational type if for any irreducible component X i

of X, there exists uniquely a connected component X 

i
of X  such that the

X Dnsqp(X •) the category of quasi-projective resolutions

Remarks (i) A birational proper morphism f : X  → X has the lifting

property Indeed, according to Hironaka [18], there exists a variety X together

with morphisms g : X  → X and h : X  → X  , such that f h = g and

g is obtained by iterating blowing-ups with smooth centers (Here we may

assume that the centers are smooth using Hironaka’s theory of resolution ofsingularities.) This implies that a proper morphism has the lifting property ifthe generic points of the irreducible components can be lifted

(ii) For X • ∈ C b, ≥j

X and a closed subcomplex Y • such that dim Y • < dim X •,

there exists a smooth quasi-projective modification (X 

• , Y 

•)→ (X • , Y •) of gree ≥ j by replacing Y • with a larger subcomplex of the same dimension if

de-necessary This follows from [17, I, 2.6], except the birationality of X 

j → X j,

because there are connected components of X 

• which are not birational to

ir-reducible components of X • Indeed, if we denote by Z i,a the images of the

irreducible components of X k (k ≤ i) by morphisms to X i which are obtained

by composing morphisms appearing in the differential of X •, then the

con-nected components of X 

i are ‘sufficiently blown-up’ resolutions of singularities

of Z i,a , and are defined by increasing induction on i, lifting the differential of X •

(see loc cit ) However, if Z j,a is a proper closed subvariety of some irreducible

component X j,b of X j , we may replace the resolution of Z j,aby its lifting to the

resolution of X j,b using the lifting property, because the differential X j → X j −1

is zero

2.2 Proposition For any X • ∈ C b, ≥j

X , there exists a quasi-projective

X Dnsqp and for any irreducible component Z j,i of Z j

the restriction of the differential to some irreducible component Z j+1,iof Z j+1

is given by an isomorphism onto Z j,i with coefficient ±1, under the inductive

hypothesis:

(2.2.2) Xj+1 →  X j has the lifting property.

Indeed, admitting this, Z • is then isomorphic to the mapping cone of

j,i →  X j,i up to a sign If dim X 

j,i = dim X • , then f i

induces a birational morphism to Im f iand we may assume that there exists anirreducible component X 

j+1,i such that the restriction of d to  X 

differential d of  X 

• is defined by lifting d of  X • (see loc cit ) Then we can

modify the morphism X 

j+1 →  X j+1 by using f i for dim X j,i < dim  X •, andreplace X 

j with the union of the maximal dimensional components So wemay assume that X 

j is equidimensional, because the modified X 

• →  X • stillinduces an isomorphism over the complement of Y •by replacing Y •if necessary.Here we may assume also that Y j+1 →  Y j has the lifting property by taking



Y •appropriately (due to (2.2.2) and the above Remark (i)) Then, consideringthe mapping cone of Y 

• →  Y •, the first assertion follows by induction

The proof of the second assertion is similar Consider the shifted mappingcone (i.e the first term has degree zero):

• = [X 1, • ⊕ X 2, • → X • ], where the morphism is given by u1− u2 Then X 

• → X a, • is a strong

quasi-isomorphism Note that the composition of the canonical morphism X 

• → X a, •

and u a is independent of a up to homotopy.

By definition, for any irreducible component X 

j −1,iwhich have

the lifting property (with coefficient±1) Then by the same argument as above,

we have a smooth quasi-projective modification u  : (X 

• , Y 

•) → (X 

• , Y 

•) ofdegree ≥ j − 1 Here we may assume that the connected component X 

j −1,i , and X j  has two connected components such that the

restriction of d to each of these components is given by an isomorphism onto

X 

j −1,i (with coefficients ±1) We may also assume that Y 

j → Y 

j −1 has the

lifting property as before

Then, applying the same argument to C(Y 

• → Y 

•), and using induction

on dimension, we get a strong quasi-isomorphism

j) to these components are given by isomorphisms onto X j −1,i

(resp by birational proper morphisms to Z i , Z 

i) with coefficients ±1 Thus

Remark It is not clear if for any u i ∈ K b, ≥j

X Dnsqp(X • ) (i = 1, 2) and

v a : u2 → u1, there exists u3 ∈ K b, ≥j

X Dnsqp(X • ) together with w : u3 → u2

such that v1w = v2w This condition is necessary to define an inductive limit

over the category K b, ≥j

X Dnsqp(X •) If we drop the condition on the degree ≥ j,

it can be proved for K b

X Dnsqp(X •) by using the mapping cone (2.2.3)

In-deed, let K i, • be the source of u i for i = 1, 2, and K 3, • the mapping cone of

(v1 − v2, 0) : C(K 2, • → 0) → C(K 1, • → K •)[−2], choosing a homotopy

h such that dh + hd = u1v1 − u1v2 Then w : K 3, • → K 2, • is given by

the projection, and v1w − v2w : K 3, • → K 1, • factors through a morphism

(v1− v2, 0) : K 3, • → C(K 1, • → K •)[−1], which is homotopic to zero.

2.3 Corollary For a complex algebraic variety X and a closed

subva-riety Y , there is a canonical integral weight filtration W on the relative mology H j (X, Y ; Z) Furthermore, it is defined by a quasi-projective resolution

coho-X •

X, Y is the closure of Y in X, and D = X \ X.

Remarks (i) The first assertion is due to Gillet and Soul´e ([14, 3.1.2]) in

the case X is proper (replacing X • with a simplicial resolution) It is expectedthat their integral weight filtration coincides with ours

(ii) If X is proper, we have

(2.3.1) W i −1 H i (X, Y ; Z) = Ker(H i (X, Y ; Z) → H i (X  ,Z))

for any resolution of singularities X  → X (see also loc cit.) Note that

π ∗ : H i (X  , Z) → H i (X  ,Z) is injective for any birational proper morphism of

smooth varieties π : X  → X .

Proof of (2.3) The canonical mixed Hodge structure on the relative

co-homology can also be defined by using any quasi-projective resolution X • → C(Y → X) as in [10] This gives an integral weight filtration together with an

integral weight spectral sequence

(2.3.2) E1p,q= ⊕ k ≥0 H q −2k( D p+k k , Z)(−k) ⇒ H p+q (X • , Z) = H p+q (X, Y ; Z),

where D j k the disjoint union of the intersections of k irreducible components

of D j, and the cohomology is defined by taking the canonical flasque resolution

of Godement in the analytic or Zariski topology By (2.2) we get a set of

integral weight filtrations on H j (X, Y ;Z) which is directed with respect tothe natural ordering by the inclusion relation Then this is stationary by the

noetherian property (It is constant if X is proper; see (2.5) below.) By the proof of (2.2) the limit is independent of the choice of the compactification X.

So the assertion follows

2.4 Definition For a complex of k-varieties X • (see (2.1)), we define

(2.4.1) Pic(X • ) = H1(X • , O ∗

X • ) (see [2]).

The right-hand side is defined by taking the canonical flasque resolution of

Godement which is compatible with the pull-back by the differential of X • For

a k-variety X and closed subvariety Y , we define the derived relative Picard

groups by

(2.4.2) Pic(X, Y ; i) = lim

−→ Pic(X • [i]), where the inductive limit is taken over X • ∈ K b

Xnsqp(C(Y → X)) If Y is empty,

Pic(X, Y ; i) will be denoted by Pic(X, i), and i will be omitted if i = 0.

Remark We can define similarly the derived relative Chow cohomology

group by

(2.4.3) CHp (X, Y ; i) = lim −→ H p+i (X • , K p ),

whereK p is the Zariski sheafification of Quillen’s higher K-group (In the case

X is smooth proper and Y is empty, this is related to Bloch’s higher Chow

group for i = 0, −1.)

The following is a variant of a result of Gillet and Soul´e [14, 3.1], and gives

a positive answer to the question in [2, 4.4.4]

2.5 Proposition Assume X, Y proper Then a strong quasi

where we assume k = C for the second morphism In particular, the inductive

system in (2.4.2) is a constant system in this case.

Proof It is sufficient to show that Pic(X • [i]) = 0 and the E1-complex

ZE1• ,q of the integral weight spectral sequence is acyclic, if X •is strongly acyclic

and the X j are smooth (Note that the E1-complex for W is compatible with the mapping cone.) Considering the E1-complex of the spectral sequence

(2.5.1) P E1p,q = H q (X p , O ∗

X p)⇒ H p+q (X • , O ∗

X • ),

it is enough to show the acyclicity of the complexes P E1• ,q and ZE1• ,q (where

P E1• ,q = 0 for q > 1; see (2.5.3) below).

By Gillet and Soul´e ([14, 1.2]) this is further reduced to the acyclicity

of the Gersten complex of X • × V for any smooth proper variety V because

it implies that the image of the complex X • in the category of complexes

of varieties whose differentials and morphisms are given by correspondences ishomotopic to zero Since the functor associating the Gersten complex preserveshomotopy, it is sufficient to show that the Gersten complex of

Z  → Z ⊕ Y  → Y

is acyclic in the notation of (2.1.1) (replacing it by the product with V ) sider the subcomplex of the Gersten complex given by the points of Z  , Z,

Con-Y  , Y contained in Z  , Z, Z  , Z respectively It is clearly acyclic, and so is

its quotient complex This shows the desired assertion (A similar argumentworks also for (2.4.3).)

Remark Let X be a smooth irreducible k-variety, and k(X) the function

field of X For closed subvariety D, let k(X) ∗

X and ZD denote the constant

sheaf in the Zariski topology on X and D with stalk k(X) ∗ andZ respectively.Then we have a flasque resolution

for an open immersion j : U → X.

2.6 Proposition There is a canonical long exact sequence

(2.6.1) → Pic(X, Y ; i) → Pic(X, i) → Pic(Y, i) → Pic(X, Y ; i + 1) →

Remark This is an analogue of the localization sequence for higher Chow

X , because for any quasi-projective resolutions u : X • → X and

v : Y • → Y , there exists quasi-isomorphisms u  : X 

2.7 Remark Assume k = C and X is proper Let H i+2

D (X, Y ;Z(1))denote the relative Deligne cohomology See [3] and also (5.2) below Then wecan show

(2.7.1) Pic(X, Y ; i) = H D i+2 (X, Y ; Z(1)) for i ≤ 0.

This is analogous to the canonical isomorphisms for

CH1(X, i) = H 2nAH−2+i (X, Z(n − 1)) which holds for i > 0 and any variety X of dimension n [31].

Assume X is proper and normal Let H1

XZ(1) be the Zariski sheaf

asso-ciated with the presheaf U → H1(U,Z(1)) By the Leray spectral sequence we

get a natural injective morphism HZar1 (X, H1

Let X • → X be a quasi-projective resolution Then

(2.7.3) H D2(X,Z(1))alg = Im(Pic(X) → Pic(X • ) = H D2(X, Z(1))).

Indeed, if we put NS(X • ) := Im(Pic(X •) → H2(X,Z(1))), and similarly for

NS(X), this follows from a result of Biswas and Srinivas [6]:

phisms of H1(X, C) to the source and the target of (∗)).

3 Construction

We construct 1-motives associated with a complex of varieties, and show

the compatibility for the l-adic and de Rham realizations We assume k is an

algebraically closed field of characteristic zero

3.1 With the notation of (2.1), let X • ∈ C k be a complex of smooth

k-varieties (see (2.1)), and X • a smooth compactification of X • such that

D p := X p \ X p is a divisor with simple normal crossings We assume X • is

bounded below The reader can also assume that (X • , D •) is the mapping

to be the quotient complexes of X • consisting of X i for i ≥ p (and empty

otherwise) This is similar to the filtration “bˆete” σ in [10] It induces a decreasing filtration W  on K such that

W

K is quasi-isomorphic to Γ(  D • ,Z), and Γ( D p , Z)  Z ⊕r p, where D •

is the normalization of D • , and r p is the number of irreducible components of

D p (Indeed, a constant sheaf on an irreducible variety is flasque in the Zariskitopology.)

Let W be the convolution of W  and W  (i.e., W r=

X r ) is the Picard group of X r.LetK  = Gr0

W

K (= Γ(X • , C •(O ∗

X • ))) with the induced filtration W Then

we have the spectral sequence

(3.1.3) E1p,q =

H q (X p , O ∗

X p) for p ≥ r

0 for p < r ⇒ H p+q (W r K  ), which degenerates at E3 We define

variety (This is well-known for P r (X • ).) Let P ≥r (X •)0 denote the identity

component of P ≥r (X • ) This is identified with P ≥r (X • • )0 by the above

exact sequence (and similarly for P r (X0• ), P r (X • • )0)

By the boundary map ∂ of the long exact sequence associated with

Let Γ

r (X • • ) be the kernel of the right vertical morphism of (3.1.4) Put

NS(X • • ) j := P j (X • • )/P j (X •)0 = NS(X j)⊕ Γ(  D j+1 , Z),

where

NS(X j ) := Pic(X j )/Pic(X j)0 = HomMHS(Z, H2(X j , Z)(1)).

Then (NS(X • • ) • , d ∗) is the single complex associated with a double complex

such that one of the differentials is the Gysin morphism

r (X • • ) are identified with locally finite commutative

group schemes Then (3.1.4) induces morphisms

(This construction is equivalent to the one in [26].)

Remark By (3.1.3), P ≥r (X •) is identified with the group of isomorphism

classes of (L, γ) where L is a line bundle on X r and

γ : O X r+1 → d ∼ ∗ L

is a trivialization such that

d ∗ γ : d ∗ O X r+1(=O X r+2)→ (d ∼ 2)∗ L = O X r+2

is the identity morphism (Note that d ∗ L is defined by using tensor of line

bundles.) See also [2], [26]

For the construction of the group scheme P ≥r (X •), we need dieck’s theory of representable group functors (see [15], [23]) as follows:3.2 Theorem There exists a k-group scheme locally of finite type

Grothen-P ≥r (X • ) such that the group of its k-valued points is isomorphic to P ≥r (X • ).

Moreover, P ≥r (X • ) has the following universal property: For any k-variety S